On Pairing Observations from a Distribution with Monotone Likelihood Ratio

Abstract

It is assumed that a random sample of size $n$ is taken from a bivariate distribution whose density $f(x, y)$ possesses a Monotone Likelihood Ratio, i.e. for all $x_1 < x_2$ and $y_1 < y_2, f(x_1, y_1)f(x_2, y_2) \geqq f(x_1, y_2)f(x_2, y_1)$. When the sample is "broken," i.e. when the $x$- and $y$-values are received in random relative order, it is desirable to optimally "reconstruct" the original bivariate sample. Optimal properties of the Maximum Likelihood Pairing (MLP) of $x$- and $y$-values, obtained by DeGroot, et al. in [1], are generalized to the class of distributions defined above, with particular attention given to the trinomial distribution. In addition, one of the main results shown is that in general the MLP is better than random pairing, in that the expected number of correct pairings using the MLP is greater than unity.